1. The Problem
You are to design a high boost filter using the windowing filter design method and apply it to an audio signal. The desired discrete system frequency response is shown in Fig. 1. Create an FIR filter with M = 1001 impulse response samples. Define the filter impulse response samples analytically, by following the windowing filter design method. Evaluate the equations in MATLAB to get your impulse response. You may use freqz( ) to plot your frequency response, but you should not use fdatool( ), or fir1( ) or any other ready-made filter design function. Use the rectangular window and another window of your choosing. Process a short audio signal (of your choosing) and listen to the input and output using the sound( ) or soundsc( ) command.
Figure 1:Desired discrete system frequency response. Frequency is in radians/sample.
2. What to Include
You may use Word or LaTeX, but produce and submit a .pdf file and do not compress or archive it).
You should begin with a title page. Then using numbered section headings include the following sections: 1. Introduction, 2. Filter Design, 3. Experimental Results, and 4. Conclusions. Include an Appendix section that includes your MATLAB code. In the Introduction, explain the problem in your own words. In the Filter Design section, include the following items:
ï‚· Starting from the desired frequency response in Fig. 1. Follow the windowing filter design method to produce an analytic expression for the rectangular windowed impulse response. Show several steps with numbered equation entered with the equation editor. Center the equations and right justify the equation numbers. Put your equation in properly punctuated complete sentences. Make sure you include the final impulse response equation.
ï‚· Include a stem plot of the impulse response with rectangular window and Blackman-Harris window.
ï‚· Show the magnitude frequency response for your two filters (put both curves on one plot and include a legend). You may use ffts, freqz, or evaluate samples of the DTFT of your impulse response. Be sure to correctly label your axes and include proper units.
4. Write Up
All figures should have a numbered caption with brief description and also be referred to in the body of the text. That is, introduce each figure by number and explain it in the section text (not just the caption). For example, in the filter design section, you might have a sentence like this: â€œFigure 2 shows the frequency response of the filter. Note that…â€ Also, all equations should be part of complete sentences with proper punctuation. That means you would put a period at the end of a sentence even it ends with an equation. Capitalize with word â€œfigureâ€ if referring to a specific one by number. Do the same for â€œequationâ€ and â€œsectionâ€, etc.
Include all MATLAB code used for the project in the appendix. Be sure the code is clear, concise, and well commented. It should run without error if copied and pasted into the command window.
Make sure to include all of the results listed above. The body of your text should be
contiguous (approximately 1-2 pages of text). Embed figures or include them at the end.
The body of the text should refer to each figure by number and explain it clearly.
1.The first step for designing a FIR digital filter using the windowing method is to specify the frequency response of this filter. In this question, the desired discrete frequency response is explained in figure (2).
Figure 2: Desired discrete system frequency response. Frequency is in radians/sample.
2.The second step is to determine the inverse of the Discrete Time Fourier Transform IDTFT of as following.
3. Windowing the impulse response of the filter as following:
Where w[n] is M non-zero window function. The rectangular and the Blackman-Harris windows will be used.
For the rectangular window; (3)
The Matlab function w= window(@rectwin,M);
By multiplying with a rectangular window (to make FIR)
3.Shifting the windowing impulse response by (M-1)/2 if the causality is necessary. In this design M=1001.
4. Blackman-Harris window
The equation Blackman-Harris window is ,
where N is the window length.
Only you need to use the Matlab function w= window(@blackmanharris,M);
M = 1001;
wc = pi/4;